A generalized theory of fracture mechanics

A fracture-mechanics theory is developed which gives fracture criteria for solids in general, without limitations as to their linearity, elastic behaviour or infinitesimal strain. Besides the “standard” results of the theory which reduce to familiar forms like the Griffith equation for linear, elastic solids, several new results emerge from the theory. These include a relationship between the surface work and the true surface energy of the solid, an explanation of certain departures from standard fracture mechanics obtained with inelastic materials, and a prediction and explanation of the phenomenon of notch brittleness. Further applications of the theory, such as adhesive failure and fatigue, will be explored in a subsequent paper.     

A linear asymptotic theory for anisotropic shells

Summary In the standard treatments for anisotropic shells based on the Kirchoff hypothesis, it is necessary to make certain restrictions on the type of anisotropy and the resulting theories involve only six elastic constants.In the present work, a shell theory is obtained by an asymptotic or perturbation method which does not require any restriction on the anisotropy.It is found that, in those cases in which the extensional and bending strains are of the same order of magnitude, the leading terms satisfy the classical equations and depend only on the same six elastic constants.It is seen however that in some cases the full anisotropy is significant and it is shown that in the extension of a plate the anisotropy can produce displacements normal to the plate.     

Crack stability in linear elastic fracture mechanics

A linear elastic fracture mechanics analysis of the conditions that produce crack instability is presented. If the initial crack extension causes the change in crack-tip resistance to be negative with respect to the change in crack-tip loading, the crack will continue to propagate even though the loading agent remains stationary, and the crack is defined as unstable. The value of crack-tip load when the crack becomes unstable, G _c, is not only a function of the plate material and thickness and fracture mode, but also depends on the specimen geometry and size, and on the compliance of the loading system. The crack-tip resistance, G _R, on the other hand, is essentially a property of the plate material and thickness and fracture mode if the crack-propagation is time independent. Once G _R has been experimentally determined as a function of crack-propagation distance for a particular plate material and thickness and fracture mode, the value of G _e can be calculated for the same material and thickness and fracture mode for any plate configuration for which the elastic stress analysis is known.

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Zusammenfassung In dem Gebiet der Mechanik elastischer linearer Risse ist eine Analyse der Bedingungen, die zur Sprungbestdndigkeit führen, dargestellt. Wenn die ursprüngliche Sprungverlängerung veranlaßt, daß sick die Veranderung in dem Widerstand des Sprungendes in Bezug auf die Veranderung in der Belastung des Sprungendes negativ verhält, wird der Sprung sich weiter verlängern, obwohl das Belastungsmittel stillsteht, and der Sprung wird als widerstandsunfähig bezeichnet. Der Wert der Sprungendenbelastung wenn der Sprung widerstandsunfäbig wird, G _c, ist nicht nur eine Funktion des Plattenmaterials, des Plattendicke and dert Art des Risses, sondern hängt auch von der Geometric und der Größe des Stückes, Bowie von der Anpassung des Belastungssystems ab. Wenn, andererseits, die Sprungvergrößerung unabhängig von der Zeit vor sich geht, so ist die Sprungendenfestigkeit, G _R, im wesentlichen die Eigenschaft des Plattenmaterials, des Plattendicke and der Art des Risses. Wenn G _R erst einmal durch Versuche als Funktion der Länge der Sprungvergrößerung für ein bestimmtes Plattenmaterial, Plattendicke and RiBart bestimmt ist, so kann der Wert von G _c für das selbe Material, Dicke and Rißart in jeder Plattengeometrie, für welche die Analyse der elastischen Beanspruchung bekannt ist, bezechnet werden.

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Résumé On rapporte l'analyse des conditions produisant l'instabilité des fissures. Au cas où le prolongement de la fissure initiale modifie la résistance de la pointe de la criqûre à une valeur négative, au regard du changement en charge de la pointe, la fissure continuera à se propager même quand le système de charge reste stationnaire. Dans ce cas, la fissure est considéré comme instable. La valeur de la charge de la pointe d'une fissure instable (G _c) n'est pas seulement une caractéristique du matériau du panneau mais dépend en plus de la géometrie de l'échantillon et de ses dimensions ainsi que de la complaisance du système de charge. D'autre part, la résistance de la pointe de la fissure (G _R) est essentiellement une caractéristique du matériau du panneau et de son épaisseur ainsi que de son mode de rupture si la propagation de la criqure est indépendant du temps. Autant que l'on aura déterminé G _R, par l'expérience, en fonction de la distance de propagation de la fissure pour un matériau de panneau donné et de son épaisseur ainsi que de son mode de rupture, la valcur de G _c pent être calculé, pour un même matériau de même épaisseur et mode de rupture, pour toute configuration en panneau dont on sait l'analyse élastique.

     

Linear elastic fracture mechanics applied to cracked plates and shells

A general theory concerning through-cracks in plates and shells is proposed. Applying the method of local development of the kinematic unknowns to a shell of arbitrary shape, the distribution of displacements and rotations as functions of polar coordinates in the vicinity of the crack tip is given and the five corresponding intensity factors are defined. New path-independent integrals are introduced and related to the intensity factors so that these can be evaluated numerically. Finally, a fracture criterion for plane problems is extended to shells.

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Résumé On propose une théorie générale du comportement des fissures traversantes dans les plaques et les coques. Pour cela, on applique la méthode du développement limité des variables cinématiques à une coque de forme quelconque. On obtient la distribution des déplacements et des rotations en fonction des coordonnées polaires dans le voisinage de la singularité et on définit les cinq facteurs d'intensité correspondants. On propose de nouvelles intégrales curvilignes independantes du contour et on les relie aux facteurs d'intensité de telle sorte que ces derniers peuvent être calculés numériquement. Enfin on étend aux coques un critère de rupture utilisé jusqu'alors dans les problèmes plans.

     

Limitations of elastic definitions in Al-Si-Mg cast alloys with enhanced plasticity: linear elastic fracture mechanics versus elastic-plastic fracture mechanics

Linear elastic fracture mechanics describes the fracture behavior of materials and components that respond elastically under loading. This approach is valuable and accurate for the continuum analysis of crack growth in brittle and high strength materials; however it introduces increasing inaccuracies for low-strength/high-ductility alloys (particularly low-carbon steels and light metal alloys). In the case of ductile alloys, different degrees of plastic deformation precede and accompany crack initiation and propagation, and a non-linear ductile fracture mechanics approach better characterizes the fatigue and fracture behavior under elastic-plastic conditions.To delineate plasticity effects in upper Region II and Region III of crack growth an analysis comparing linear elastic stress intensity factor ranges (ΔK_el) with crack tip plasticity adjusted linear elastic stress intensity factor ranges (ΔK_pl) is presented. To compute plasticity corrected stress intensity factor ranges (ΔK_pl), a new relationship for plastic zone size determination was developed taking into account effects of plane-strain and plane-stress conditions (“combo plastic zone”). In addition, for the upper part of the fatigue crack growth curve, elastic-plastic (cyclic J based) stress intensity factor ranges (ΔK_J) were computed from load-displacement records and compared to plasticity corrected stress intensity factor ranges (ΔK_pl). A new cyclic J analysis was designed to compute elastic-plastic stress intensity factor ranges (ΔK_J) by determining cumulative plastic damage from load-displacement records captured in load-control (K-control) fatigue crack growth tests. The cyclic J analysis provides the true fatigue crack growth behavior of the material. A methodology to evaluate the lower and upper bound fracture toughness of the material (J_IC and J_max) directly from fatigue crack growth test data (ΔK_FT(JIC) and ΔK_FT(Jmax)) was developed and validated using static fracture toughness test results. The value of ΔK_FT(JIC) (and implicitly J_IC) is determined by comparing the plasticity corrected elastic fatigue crack growth curve with the elastic-plastic fatigue crack growth curve. A most relevant finding is that plasticity adjusted linear elastic stress intensity factor ranges (ΔK_pl) are in remarkably good agreement with cyclic J analysis results (ΔK_J), and provide accurate plasticity corrections up to a ΔK corresponding to J_IC (i.e. ΔK_FT(JIC)). Towards the end of the fatigue crack growth test (above ΔK_FT(JIC)) when plasticity is accompanied by significant tearing, the cyclic J analysis provides a more accurate way to capture the true behavior of the material and determine ΔK_FT(Jmax). A procedure to decouple and partition plasticity and tearing effects on crack growth rates is given.Three cast Al-Si-Mg alloys with different levels of ductility, provided by different Si contents and heat treatments (T61 and T4) are evaluated, and the effects of crack tip plasticity on fatigue crack growth are assessed. Fatigue crack growth tests were conducted at a constant stress ratio, R = 0.1, using compact tension specimens.     

Griffith theory and development of fracture mechanics criteria

A. A. Blagonravov Institute of Engineering, Russian Academy of Sciences, Moscow. Translated from Fiziko-Khimicheskya Mekhnika, Vol. 29, No. 3, pp. 140–145, May–June, 1993.     

From Griffith's theory to the fractal fracture mechanics

Conclusion The above analysis shows the necessity for further development of fractal fracture mechanics at micro-, meso-, and macrolevels, using fractal theory and the general principle of synergetics.A. A. Baikov Institute of Metallurgy, Russian Academy of Sciences, Moscow. Published in Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 29, No. 3, pp. 101–106, May–June, 1993.     

Weakly nonlinear fracture mechanics: experiments and theory

Material failure occurs at the small scales in the immediate vicinity of the tip of a crack. Due to its generally microscopic size and the typically high crack propagation velocity, direct observation of the dynamic behavior in this highly deformed region has been prohibitively difficult. Here we present direct measurements of the deformation surrounding the tip of dynamic mode I cracks propagating in brittle elastomers at velocities ranging from 0.2 to 0.8C _s . Both the detailed fracture dynamics and fractography of these materials are identical to that of standard brittle amorphous materials such as soda-lime glass. These measurements demonstrate how Linear Elastic Fracture Mechanics (LEFM) breaks down near the tip of a crack. This breakdown is quantitatively described by extending LEFM to the weakly nonlinear regime, by considering nonlinear elastic constitutive laws up to second order in the displacement-gradients. The theory predicts that, at scales within a dynamic lengthscale ? _nl from the tip of a single crack, significant logr displacements and 1/r displacement-gradient contributions arise, and provides excellent quantitative agreement with the measured near-tip deformation. As ? _nl is consistent with lengthscales that appear in crack tip instabilities, this “weakly nonlinear fracture mechanics” framework may serve as a springboard for the development of a comprehensive theory of fracture dynamics.     

New path independent integrals in linear elastic fracture mechanics

In this paper the properties of eigenfunction expansion form (abbreviated as EEF) in the crack problems of plane elasticity and antiplane elasticity are discussed in details. After using the Betti's reciprocal theorem to the cracked body, several new path independent integrals are obtained. All the coefficients in the EEF at the crack-tip, including the K₁, K₂ and k₃ values, can be related to corresponding path-independent integrals.     

A reappraisal of transition elements in linear elastic fracture mechanics

This article offers a detailed comparison of the transition elements described by P.P. Lynn and A.R. Ingraffea [International Journal for Numerical Methods in Engineering 12,1031–1036] and C. Manu[Engineering Fracture Mechanics 24,509–512]. The source of a numerical phenomenon in using Manu's transitionelement (TE) is explained. The effect of eight-noded TEs with differentquarter-point elements (QPE) on the calculated stress intensity factors (SIFs) isinvestigated. Strain at the crack tip is shown to be singular for any ray emanating from the crack tip within an eight-noded TE, but strain has bothr ^–1/2andr ^–1singularities, withr ^–1/2dominating for large TEs. Semi-transition elements (STEs) are defined and shown to have a marginal effect on the calculated SIFs. Nine-nodedtransition elements are formulated whose strain singularity is shown to be the same as that of eight-noded TEs. Then the effect of eight-noded and nine-noded TEs with collapsed triangular QPEs, and rectangular and nonrectangular quadrilateral eight-noded and nine-noded QPEs, is studied, and nine-noded TEs are shown to behave exactly like eight-noded TEs with rectangular eight-noded and nine-noded QPEs and to behave almost the same with other QPEs. The layered transition elements proposed by V. Murti and S.Valliapan [Engineering Fracture Mechanics 25, 237–258] areformulated correctly. The effect of layered transition elements is shown by two numerical examples.     

The use of linear elastic fracture mechanics for particulate solids

The fracture of model porous particulate solids consisting of sand and low volume fractions of a polymeric binder has been investigated. In order to apply a linear elastic fracture mechanics analysis it was necessary to use an effective crack length comprising the sum of the notch depth and a quantity a. The behaviour of the model solids was intermediate between that exhibited by ceramics and polymers. A possible interpretation of a is that it represents a process zone size where the energy dissipation mechanism could involve either microcracking or yielding of the interparticle junctions in the zone. The former would correspond to ceramic-like behaviour while the latter is characteristic of polymers.     

On the relevance of linear elastic fracture mechanics to ice

A new criterion is proposed which allows one to estimate the minimum size for a brittle ice fracture specimen comprised of a small number of grains. According to this criterion, linear elastic fracture mechanics is a useful theory for fresh-water ice but may have limited use for saline ice.